Rodion Kuzmin

Template:Infobox scientist Rodion Osievich Kuzmin (Russian: Родион Осиевич Кузьмин{{#invoke:Category handler|main}}, Nov. 9, 1891, Riabye village in the Haradok district – March 23, 1949, Leningrad) was a Russian mathematician, known for his works in number theory and analysis. His name is sometimes transliterated as Kusmin.

Selected results

• In 1928, Kuzmin solved the following problem due to Gauss (see Gauss–Kuzmin distribution): if x is a random number chosen uniformly in (0, 1), and

$x={\frac {1}{k_{1}+{\frac {1}{k_{2}+\cdots }}}}$

is its continued fraction expansion, find a bound for

$\Delta _{n}(s)=\mathbb {P} \left\{x_{n}\leq s\right\}-\log _{2}(1+s),$

where

$x_{n}={\frac {1}{k_{n+1}+{\frac {1}{k_{n+2}+\cdots }}}}.$

Gauss showed that Δ_n tends to zero as n goes to infinity, however, he was unable to give an explicit bound. Kuzmin showed that

$|\Delta _{n}(s)|\leq Ce^{-\alpha {\sqrt {n}}}~,$

where C,α > 0 are numerical constants. In 1929, the bound was improved to C 0.7ⁿ by Paul Lévy.

• In 1930, Kuzmin proved that numbers of the form a^b, where a is algebraic and b is a real quadratic irrational, are transcendental. In particular, this result implies that Gelfond–Schneider constant

$2^{\sqrt {2}}=2.6651441426902251886502972498731\ldots$

is transcendental. See Gelfond–Schneider theorem for later developments.

• He is also known for the Kusmin-Landau inequality: If $f$ is continuously differentiable with monotonic derivative $f'$ satisfying $\Vert f'(x)\Vert \geq \lambda >0$ (where $\Vert \cdot \Vert$ denotes the Nearest integer function) on a finite interval $I$, then

$\sum _{n\in I}e^{2\pi if(n)}\ll \lambda ^{-1}.$

External links

• Rodion Kuzmin at the Mathematics Genealogy Project (The chronology there is apparently wrong, since J. V. Uspensky lived in USA from 1929.)